where \( \mathcal {M}_e(X)\)  is the set of all \(\sigma \) -ergodic measures supported by \(X\)  and we denote by \(h_{\mu ,p,\mathbf {I}}=h_{\mu ,V,\mathbf {I}}\)  for \(V\)  in (REF ). By the proof of [1]}, one can find an ergodic measure \(\mu \)  supported by a \(\sigma ^k\) -invariant subset \(y\)  of \(X\)  for some large integer \(k\)  such that the \(k\) -iterated IFS \(\mathbf {I}^k\)  restricted on \(y\)  satisfies the strong separation condition, hence \(h_{p,\mu ,\mathbf {I}^k}=0\) ; and its dimension through the canonical mapping \(\Phi _{\mathbf {I}}\)  can be arbitrarily close to \(\overline{\dim }_B X_{\mathbf {I}}\) . Therefore, by (REF ) we obtain a lower bound for \(\underline{\dim }_H X_{\mathbf {I},p}\) :
\(\underline{\dim }_H X_{\mathbf {I},p}\ge \overline{\dim }_B X_{\mathbf {I}}- \frac{\log p}{\log \delta }.\) 
